non-euclidean geometry

Thân thiện

A student draws a diagram of non-euclidean geometry on a chalkboard.

Definition

1. Noun:
   • A system of geometry: A branch of mathematics that is based on a set of axioms (fundamental rules) different from those established by the ancient Greek mathematician Euclid. It discards or modifies at least one of Euclid's traditional postulates, particularly the famous parallel postulate, leading to a logically consistent but different description of space.

Usage Examples

• Noun:
  • The development of non-Euclidean geometry in the 19th century revolutionized our understanding of mathematical space.
  • In non-Euclidean geometry, the angles of a triangle do not necessarily sum to 180 degrees.
  • Einstein's theory of general relativity uses the framework of non-Euclidean geometry to describe gravity.

Advanced Usage

• "to work in/with non-Euclidean geometry": To conduct research or calculations using the principles of this geometry.
  • The physicist had to learn to work in non-Euclidean geometry to develop the new model.
• As a conceptual metaphor: Used to describe a situation or system operating under a fundamentally different set of rules.
  • Navigating the corporate bureaucracy felt like applying non-Euclidean geometry to a simple problem.

Variants and Related Words

• Hyperbolic geometry (n): A specific type of non-Euclidean geometry where, through a point not on a given line, there are lines parallel to the given line.
• Elliptic geometry (n): A specific type of non-Euclidean geometry where there are parallel lines (e.g., geometry on the surface of a sphere).
• Riemannian geometry (n): A broad, differential form of non-Euclidean geometry foundational to general relativity.

Synonyms

• Non-Euclidean (adj): The adjective form describing concepts or spaces governed by these rules (e.g., non-Euclidean space).
• There is no single-word synonym for this specific, well-defined mathematical term.

Related Phrases

• Parallel postulate: The specific Euclidean axiom concerning parallel lines that is altered or rejected in non-Euclidean systems.
• Curved space: A common description of the space described by non-Euclidean geometry, in contrast to the "flat" space of Euclidean geometry.

Học thuật

Noun

1. (mathematics) geometry based on axioms different from Euclid's
   • non-Euclidean geometries discard or replace one or more of the Euclidean axioms

Từ đồng nghĩa

• non-euclidean geometry